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On Matveev-Piergallini moves for branched spines by Muramatsu, Kohei; Suzuki, Sakie; Taguchi, Koki is a scholarly article available to read on EtoBox.

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The Matveev-Piergallini (MP) moves on spines of $3$-manifolds are well-known for their correspondence to the Pachner $2$-$3$ moves in dual ideal triangulations. Benedetti and Petronio introduced combinatorial descriptions of closed $3$-manifolds and combed $3$-manifolds by using branched spines and their equivalence relations, which involves MP moves with 16 distinct patterns of branchings. In this paper, we demonstrate that these 16 MP moves on branched spines are derived from a primary MP move, pure sliding moves and their inverses. Consequently, we obtain alternative generating sets for the equivalence relations on branched spines for closed $3$-manifolds and combed $3$-manifolds. Furthermore, we extend these results to framed $3$-manifolds and spin $3$-manifolds. These descriptions are advantageous, particularly when constructing and studying quantum invariants of links and $3$-manifolds. In various constructions of quantum invariants using (ideal) triangulations, branching structures naturally arise to facilitate the assignment of non-symmetric algebraic objects to tetrahedra. In these frameworks, the primary MP move precisely corresponds to certain algebraic pentagon relation

Author

Muramatsu, Kohei; Suzuki, Sakie; Taguchi, Koki

Published

2024

Language

EN